that models the embedding of ∞\infty-Lie algebroids into a (∞,1)-topos of ∞\infty-Lie groupoids. When restricted to ∞-Lie algebras (∞\infty-Lie algebroids over the point) the difference between the sites CAlgopCAlg^{op} and ThCartSp plays no role. In fact for that case we could just as well restrict to a site of only infinitesimal spaces, because all homs from a finite non-thickened space into an infinitesimal space are trivial anyway.

Therefor for 𝔤\mathfrak{g} and 𝔥\mathfrak{h}∞\infty-Lie algebras, a cocycle on 𝔤\mathfrak{g} with values in 𝔥\mathfrak{h} is just a morphism

Such cocycles are modeled by morphisms in dgAlgopdgAlg^{op} from a cofibrant representative of 𝔤\mathfrak{g} to a fibrant representative of 𝔥\mathfrak{h}. Since in dgAlgdgAlg all objects are fibrant, in dgAlgopdgAlg^{op} all objects are cofibrant. The cofibrant objects in the model structure on dg-algebras are the Sullivan algebras CE(𝔥)CE(\mathfrak{h}). In particular for 𝔥=bn−1ℝ\mathfrak{h} = b^{n-1}\mathbb{R} we have that CE(bn−1ℝ)CE(b^{n-1}\mathbb{R}) is a Sullivan algebra, so bn−1ℝb^{n-1} \mathbb{R} is fibrant in dgAlgopdgAlg^{op}.

We say that μ∈CE(𝔞)\mu \in CE(\mathfrak{a}) is in transgression with ω∈inv(𝔞)⊂CE(Σ𝔞)\omega \in inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) if their classes map to each other under the connecting homomorphism δ\delta:

δ:[μ]↦[ω].
\delta : [\mu] \mapsto [\omega]
\,.

The following spells out in detail how one finds to a given invariant polynomial ω\omega the cocycle that it is in transgression with.

We first regard the invariant polynomialω\omega as an element of the Weil algebraW(𝔞)W(\mathfrak{a}) under the inclusion inv(𝔞)↪W(𝔞)inv(\mathfrak{a}) \hookrightarrow W(\mathfrak{a}), where, by the very definiton of invariant polynomials, it is closed: dW(𝔞)ω=0d_{W(\mathfrak{a})} \omega = 0.

then we find an element csω∈W(𝔞)cs_\omega \in W(\mathfrak{a}) with the property that dW(𝔞)csω=ωd_{W(\mathfrak{a})} cs_\omega = \omega. This is guranteed to exist because W(𝔞)W(\mathfrak{a}) has trivial cohomology.

then we send this element csω∈W(𝔞)cs_\omega\in W(\mathfrak{a}) along the restriction map W(𝔞)→CS(𝔞)W(\mathfrak{a}) \to CS(\mathfrak{a}) to an elemeent we call ν\nu.

From the fact that all morphisms involved respect the differential and from the fact that the image of ω\omega in CE(𝔞)CE(\mathfrak{a}) vanishes it follows that

this element ν\nu satisfies dCE(𝔞)ν=0d_{CE(\mathfrak{a})} \nu = 0, hence that it is an ∞\infty-Lie algebroid cocycle.

any two different choices of csωcs_\omega lead to cocylces μ\mu that are cohomologous.

We say ν\nu is a cocycle in transgression withω\omega. We may call csωcs_{\omega} here a Chern-Simons element of ω\omega. Because for A:TX→𝔞A : T X \to \mathfrak{a} any collection of ∞-Lie algebroid valued differential forms coming dually from a dg-morphism Ω•(X)←W(𝔞):A\Omega^\bullet(X) \leftarrow W(\mathfrak{a}) : A the image ω(A)\omega(A) of ω\omega will be a curvature characteristic form and the image csω(A)cs_\omega(A) its corresponding Chern-Simons form.

In the case where 𝔤\mathfrak{g} is an ordinary semisimple Lie algebra, this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with 𝔤\mathfrak{g}-valued 1-forms. This is described in the section Semisimple Lie algebras .

The coresponding Chern-Simons elements exhibiting the transgression of these invariant polynomials give action functionals for generalized Chern-Simons theory (see the above entries for more details).

Extensions

In any (∞,1)-topos with its intrinsic notion of cohomology, a cocyclec:X→Bn+1Ac : X \to \mathbf{B}^{n+1} A classifies an extensionBnA→X^→X\mathbf{B}^n A \to \hat X \to X. This X^\hat X is nothing but the homotopy fiber of cc, or equivalently the BnA\mathbf{B}^n A-principal ∞-bundle classified by cc.

After embedding ∞-Lie algebras into the (∞,1)-topos of ∞-Lie groupoids as described above, the same abstract reasoning applies to ∞\infty-Lie algebra cocycles and the extensions of ∞\infty-Lie algebras that these classify: for c:𝔤→bnℝc : \mathfrak{g} \to b^n \mathbb{R} a cocycle of ∞\infty-Lie algebras, the extension bn−1ℝ→𝔤^→𝔤b^{n-1} \mathbb{R} \to \hat \mathfrak{g} \to \mathfrak{g} is the homotopy fiber of this morphism in ?LieGrpd.